| L(s) = 1 | + 2.41i·3-s − 1.82i·5-s + 4.41·7-s − 2.82·9-s + 0.828i·11-s − i·13-s + 4.41·15-s + 17-s − 5.65i·19-s + 10.6i·21-s + 8.82·23-s + 1.65·25-s + 0.414i·27-s − 3.65i·29-s − 3.65·31-s + ⋯ |
| L(s) = 1 | + 1.39i·3-s − 0.817i·5-s + 1.66·7-s − 0.942·9-s + 0.249i·11-s − 0.277i·13-s + 1.13·15-s + 0.242·17-s − 1.29i·19-s + 2.32i·21-s + 1.84·23-s + 0.331·25-s + 0.0797i·27-s − 0.679i·29-s − 0.656·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76873 + 0.732634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76873 + 0.732634i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 3 | \( 1 - 2.41iT - 3T^{2} \) |
| 5 | \( 1 + 1.82iT - 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 - 0.828iT - 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 5.65iT - 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65iT - 29T^{2} \) |
| 31 | \( 1 + 3.65T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 - 8.41iT - 43T^{2} \) |
| 47 | \( 1 + 0.757T + 47T^{2} \) |
| 53 | \( 1 - 3.65iT - 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 8.82iT - 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 1.65T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.34417959019268445673886022682, −9.388729861737438289164192758214, −8.774158785909572400655444073345, −8.079578868017142518496254836166, −6.96485429926486807701111239524, −5.29345826927654348898063210613, −4.91852016564932348679081484883, −4.35448492172087346691661405289, −2.97276368715446357205654987154, −1.29875759091842048829601082061,
1.28857912440002105810889353911, 2.12281371703753668064661229090, 3.44570044703651812489892767576, 4.92732403279570319434996736746, 5.83825187670412270897321407879, 6.99513501643035999530227632801, 7.35335233702109109045147029695, 8.234444976506103147235283371571, 8.929133763846230277568127731589, 10.46617040313257129805326476031
